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Math HS: Algebra I – Transformations of Functions

Introduction

This annotated task is from EngageNY materials for Algebra 1, Module 3 – Lesson 17: Four Interesting Transformations of Functions. This task is made up of the questions in a problem set at the end of the lesson. The standards targeted in this module are:

HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

HSF-IF.C.7aGraph linear and quadratic functions and show intercepts, maxima, and minima.

HSF-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

The selected task requires understanding the effect of parameters on graphs of absolute value functions and primarily targets HSF-BF.B.3. However, it also relies on and applies a foundational understanding of aspects of HAS-REI.D.11 and HSF-IF.C.7a. In this lesson, experimentation with technology and/or recognition of odd and even functions is not required (underlining indicates portions of the standards that are specifically targeted or referenced in this student task).

When the task asks for students to describe the transformation, a proficient response would specifically include (1) precise vocabulary for transformations, (2) the direction of the transformation, and (3) the quantity (distance or scale factor) involved in the transformation.

To show evidence of depth of knowledge, student work should make connections between descriptions, equations, graphs, and tables of values. Evidence of viewing transformations holistically, rather than as a point-by-point (ordered pair) transformation, also illustrates the appropriate depth of knowledge. This may not be something that is easily communicated on paper, but through class discussion and identifying errors based on the predication of the transformation, this evidence should occur.

Question #7 is a culminating task designed for the students to look for and make use of the structure (SMP.7) and repeated patterns (SMP.8) they discovered in Questions #1 through #5 and to use appropriate and precise vocabulary when describing the effects of k on the parent function (SMP.6).

In looking closely at these student work samples, we noticed that completeness and consistency is lacking in some of the samples. It might be helpful for the teacher to reinforce what the word “describe” constitutes regarding transformations, specific to effects of k on the function in each case. For example, in the transformation g(x) = f(x) + 3, students should be able to describe the transformation as a vertical shift where all the outputs in g(x) are increased by a value of 3 over those of f(x). In Question #6, correctly performing multiple transformations on the parent function shows a deeper understanding of specifically how the outputs are transformed (HSF-BF.B.3). If this were required without the scaffolding step, this would indicate “above proficient” for the requirements of the standard.

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